On SL(3,C)-representations of the Whitehead link group

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On SL(3,C)-representations of the Whitehead link group

Antonin Guilloux, Pierre Will

To cite this version:

Antonin Guilloux, Pierre Will. On SL(3,C)-representations of the Whitehead link group. Geometriae Dedicata, Springer Verlag, In press, �10.1007/s10711-018-0404-8�. �hal-01370289�

On SL(3,C)-representations of the Whitehead link group

Universit´e Grenoble Alpes Institut Fourier, B.P.74 38402 Saint-Martin-d’H`eres Cedex

France September 22, 2016

Abstract

We describe a family of representations in SL(3,C) of the fundamental group π of the Whitehead link complement. These representations are obtained by considering pairs of regular order three elements in SL(3,C) and can be seen as factorising through a quotient of π defined by a certain exceptional Dehn surgery on the Whitehead link. Our main result is that these representations form an algebraic component of the SL(3,C)-character variety of π.

1

Introduction

Let M be a manifold. The description of the character variety of π1(M ) in a Lie group G is closely

related to the study of geometric structures on M modelled on a G-space X. In this setting,

repre-sentations of π1(M ) into G appear as holonomies of (X, G)-structures. In the case of a hyperbolic

3-manifold M , a natural target group is PSL(2,C) (or SL(2,C)), as the holonomy of a hyperbolic structure on M has image contained in PSL(2,C). The study of these character varieties was ini-tiated by Thurston, in the non-compact case, who described a natural way of constructing explicit

representations of π1(M ) in PSL(2,C) using ideal triangulations of M (see [Thu]). The rough idea

is to parametrise hyperbolic ideal tetrahedra using cross-ratios, and to analyse the possible ways of constructing the hyperbolic structure on M by gluing together these ideal tetrahedra. This method gives rise to a familly of polynomial equations expressed in terms of a family of cross-ratios, which are often referred to as Thurston’s gluing equations (see Chapter 4 of [Thu]). The output of this method

is a subvariety of Cnconsisting of those tuples of parameters that satisfy Thurston’s equations, which

is called the deformation variety. Representations can be expressed in terms of the cross-ratios, and one of the main interests of the deformation variety is that it allows explicit computations, which are very precious for experiments.

Thurston’s approach has been generalized for the target groups SL(n, C) in [BFG14, GTZ15, DGG13, GGZ15]. This generalisation is geometrically meaningful. Indeed, the subgroups SU(2, 1) and SL(3, R) of SL(3, C) correspond respectively to spherical CR structures (see below) and real projective flag structures (see [FST15]), whereas SL(4, R) corresponds to projective structures on 3-manifolds, well-studied in the convex case [Ben08]. In the case of SU(2,1), the first examples following this point of view were produced by Falbel in [Fal08], who constructed and studied examples of representations of the figure 8 knot group to SU(2,1) (see also [DF15, FW14]). A parallel is also to be drawn with

higher Teichm¨uller theory in case of surfaces. In this note, we focus on the target group SL(3, C).

Though simple in spirit, this method of describing representation varieties becomes very involved when the number of tetrahedra grows. In fact, the only SL(3, C)-character variety of a hyperbolic

3-manifold that has been completely described so far with this approach is the one of the figure 8

knot complement [FGK+16], which admits an ideal triangulation by two tetrahedra (see Section 3.1

of [Thu]). Different methods have been used to describe examples of character varieties. In [HMP15],

Heusener, Mu˜noz and Porti gave another description the character variety of the figure 8 group starting

directly from the group presentation. In [MP16], Munoz and Porti described character varieties for torus knots. We will consider here the example of the Whitehead link complement (which can be triangulated by 4 ideal simplices). Denote by π its fundamental group. A possible presentation for π is:

hx, y | [x, y][x, y−1][x−1, y−1][x−1, y]i.

Let χ3(π) be the corresponding SL(3, C)-character variety, that is the GIT quotient

χ3(π) = Hom(π, SL(3, C))//SL(3, C).

The full computation of χ3(π) is not achieved as for today and seems a difficult task. Our goal here is

to describe an algebraic component of χ3(π) that contains many examples of geometrically meaningful

representations.

Our motivation comes from the study of the so-called spherical CR structures on hyperbolic

3-manifolds. These structures are examples of (G, X)-structures where X is S3 and G is PU(2,1). The

holonomy of such a structure is thus a representation of π1(M ) in PU(2,1). This motivates the study

of representations of π to SU(2, 1) which is a real form of SL(3,C). Recall that PU(2, 1) is the group

of holomorphic isometries of the complex hyperbolic plane H2

Cand SU(2, 1) is a triple cover of it. The

sphere S3 is the boundary at infinity of H2

C. In particular, spherical CR structures arise naturally

on the boundary at infinity of quotients of H2_C with non-empty discontinuity region. Spherical CR

structures can also be thought of as examples of projective flag structures on 3-manifolds on M , of

which holonomies are representations of π1(M ) to PSL(3,C).

Striking examples of spherical CR structures have been produced by R. Schwartz in [Sch01, Sch07] about fifteen years ago. There, Schwartz described what is now called a spherical CR uniformisation of the Whitehead link complement, that is a spherical CR structure with the additional property that the holonomy representation has non-empty discontinuity region with quotient homeomorphic to the Whitehead link complement (see [Der15] for a precise definition). Since then, Deraux and Falbel [DF15] produced a spherical CR uniformisation of the complement of the figure eight knot, Deraux [Der] and Acosta [Aco15] deformed this uniformisation, Deraux [Der15] described a uniformisation of

the manifold m009, and Parker-Will [PW15] described another uniformisation of the Whitehead link

complement, different from Schwartz’s one. All these examples have the common property that the image of their holonomy representation is a subgroup of PU(2, 1) generated by a pair of regular order three elements (see the introdution of [PW15] for a list).

Our goal here is to provide a common frame for all these examples. We will show that all these representations belong to a common algebraic component of the character variety of π in SL(3,C).

This component is formed by representations ρ : π −→ SL(3, C) that factor through the group π0 ₌

Z3∗ Z3. The latter group is a quotient of π, and more precisely is the fundamental group of a compact

exceptional Dehn filling of the Whitehead link complement, as we will see later on. Define X0 as the

subset of the character variety of π0 corresponding to representations generated by two regular order

three elements of SL(3, C) (recall that an order three element in SL(3, C) is regular if and only if its trace is 0).

Theorem 1. The character variety χ3(π) contains X0 as an algebraic component of dimension 4. All

representation classes in this component factor through π0.

We observe here a very similar situation to what happens in the case of the 8-knot complement. There, a 2-dimensional component of the character variety is formed by representations factorising

through a quotient. This quotient is the fundamental group of a non-hyperbolic Dehn surgery on the

8-knot complement (see [HMP15, Proposition 10.3] or [FGK+16, Section 5.3]). Moreover this quotient

is isomorphic to the (3, 3, 4)-triangle group, which is a quotient of π0. As such (see Proposition 3), the

aforementioned 2-dimensional component for the 8-knot complement is a slice of the 4-dimensional component for the Whitehead link complement described in Theorem 1.

The proof of Theorem 1 has two steps.

• First we prove that X0 is a closed Zariski subset in χ3(π) and has dimension at least 4 (see

Proposition 6).

• Secondly we consider the particular point of X0 associated to a representation ρ0 and show

that the dimension of the Zariski tangent space to χ3(π) at this point is also 4 (see Proposition

7). As a consequence, the dimension of the complex algebraic variety X0 is at most 4. The

representation ρ0 has been analysed geometrically in [PW15] and corresponds to a spherical CR

uniformisation of the Whitehead link complement.

The main technical part in our work is the proof of Proposition 7. We choose to prove this proposition using a method that is not specific to the Whitehead link complement, and we believe that it could be used to study further examples. It involves the so-called deformation variety as

described in [FGK+16]. The latter is an affine algebraic set, which is – at least around [ρ0] – a

ramified covering of the character variety. The purpose of shifting to the deformation variety is that

it allows effective computations via decorated representations and triangulations, as in [FGK+16].

In the last section, we describe an explicit family of pairwise unconjugate representations of π0, of

which conjugacy classes form a Zariski open subset of X0. This is called, in the works of Culler, Morgan

and Shalen [CS83, MS84] a tautological representation of X0 [CS83, MS84]. These representations

are defined by pairs of regular order three matrices (A, B) with no common eigenvector that are

parametrised by the traces of the four products AB, A−1B, A−1B−1 and AB−1. These parameters

are natural coordinates on X0, in view of Lawton’s description of the character variety of the rank 2

free group given in Theorem 5.

This paper is organised as follows. In Section 2, we describe the Whitehead link complement and its fundamental group from the perspective of (real) hyperbolic geometry. In particular, we gather together classical information on presentations and parabolic subgroups of π that will be needed further. Section 3 is devoted to the character variety of π. We provide basic definitions and facts on these objects. We prove Proposition 6, state Proposition 7 and derive Theorem 1 from them. In Section 4, we present the deformation variety and prove Proposition 7. Eventually, in Section 5, we

describe an explicit parametrisation of the representations in X0. The interested reader may also want

to use the companion Sage notebook [Gui] which combines the use of SnapPy [CDW] and SageMath [Dev16] to illustrate our method.

Aknowledgement: We wish to thank Miguel Acosta, Martin Deraux, Elisha Falbel and John Parker for numerous stimulating conversations. We thank Neil Hofmann, Craig Hodgson, Bruno Martelli and Carlo Petronio for kindly answering our questions.

2

Hyperbolic geometry of the Whitehead Link Complement

The Whitehead link is depicted on Figure 1. We denote by W its complement in S3 and by π the

fundamental group of W . It is a well-known fact that W carries a (unique) complete hyperbolic structure.

2.1 The ideal octahedron

The hyperbolic structure of W can be explicitely described by considering an ideal regular octahedron (Figure 1) together with identifications of its faces by hyperbolic isometries. We refer the reader to Section 3.3 of [Thu], or to Example 1 in [Wie78] for details. We briefly recall here the description of this structure which is given in [Wie78]. We will keep the notation used there.

Figure 1: The Whitehead link, and a hyperbolic regular ideal octahedron.

We denote by _{O the ideal octahedron of which vertices are given, in the upper half-space model}

of H3

R by

∞, 0, −1, −1 + i, i, −1 + i

2 .

A flattened version of this octahedron is pictured on Figure 2. We denote by u, w1and t2the isometries

of H3

Rassociated to the following elements of SL(2,C); this choice of notation is the same as in [Wie78].

u =1 i 0 1  , t2= 1 2 0 1  and w1=  1 0 −1 − i 1  . (1)

Note that t2 belongs to the group hu, w1i, since t2 = (w1−1u)2w1uw1−1u. We equip O with the face

identifications described on Figure 2. This particular choice gives a holonomy representation with image a subgroup of the Bianchi group PSL(2,Z[i]) isomorphic to the fundamental group of the Whitehead link complement. This subgroup can be seen to have index 12 in PSL(2,Z[i]) (see [Wie78]).

The octahedron from Figure 2 is a fundamental domain for the action on H3

Rof the group generated

by u and w1. Applying Poincar´e’s polyhedron theorem, one obtains the following presentation for π.

hx, y | [x, y][x, y−1][x−1, y−1][x−1, y]_i. (2)

The correspondance between (2) and the face identifications on Figure 2 is given by x _{↔ u and}

y↔ w1. Identifying projective transformations and matrices, we denote by Γ the subgroup of PSL(2,C)

generated by u and w1.

2.2 Triangulating the ideal octahedron

Connecting the vertices with coordinates_{∞ and} −1+i₂ by an edge, one obtains a decomposition of the

00 00 00 11 11 11 00 00 00 11 11 11 00 00 00 11 11 11 00 00 00 11 11 11 00 00 00 11 11 11 00000 11111 00000 11111 0 0 0 0 0 0 0 0 1 1 1 1 1 1 1 1 00000 00000 00000 00000 00000 00000 00000 00000 00000 11111 11111 11111 11111 11111 11111 11111 11111 11111 0000 0000 0000 0000 0000 0000 0000 0000 0000 1111 1111 1111 1111 1111 1111 1111 1111 1111 0000 0000 0000 0000 0000 0000 0000 0000 1111 1111 1111 1111 1111 1111 1111 1111 00000 00000 00000 00000 00000 00000 00000 00000 11111 11111 11111 11111 11111 11111 11111 11111 00000 00000 00000 00000 00000 00000 00000 00000 11111 11111 11111 11111 11111 11111 11111 11111 0000 0000 0000 0000 0000 0000 0000 0000 0000 1111 1111 1111 1111 1111 1111 1111 1111 1111 0000 0000 0000 0000 0000 0000 0000 0000 0000 1111 1111 1111 1111 1111 1111 1111 1111 1111 0000 0000 0000 0000 0000 0000 0000 0000 0000 1111 1111 1111 1111 1111 1111 1111 1111 1111 0000 0000 0000 0000 0000 0000 0000 0000 0000 1111 1111 1111 1111 1111 1111 1111 1111 1111 0000 0000 0000 0000 0000 0000 0000 0000 1111 1111 1111 1111 1111 1111 1111 1111 00000 00000 00000 00000 00000 00000 00000 00000 11111 11111 11111 11111 11111 11111 11111 11111 00000 00000 00000 00000 00000 00000 00000 00000 11111 11111 11111 11111 11111 11111 11111 11111 0000 0000 0000 0000 0000 0000 0000 0000 0000 1111 1111 1111 1111 1111 1111 1111 1111 1111 00000 00000 00000 00000 00000 00000 00000 00000 00000 11111 11111 11111 11111 11111 11111 11111 11111 11111 00000 00000 00000 00000 00000 00000 00000 00000 11111 11111 11111 11111 11111 11111 11111 11111 00000 00000 00000 00000 00000 00000 00000 00000 11111 11111 11111 11111 11111 11111 11111 11111 A B′ D′ A′ C D C′ 0 i −1 + i −i B −1+i 2 ∞ ∞ ∞ ∞ α = w1u−1 : A7−→ A0 β = w1u−1w₁−1 : B7−→ B0 γ = t−1₂ uw−1₁ : C7−→ C0 δ = u−1 : D_{7−→ D}0,

Figure 2: An octahedron with face identifications

∆0=  i, 0,−1 + i 2 ,∞  ∆1 =  −1 + i, i,−1 + i₂ ,_∞ ∆2=  −i, −1 + i,−1 + i 2 ,∞  ∆3 =  0,−i,−1 + i 2 ,∞  (3)

This gives an ideal triangulation of the manifold W which is the one used by the software SnapPy [CDW] to provide another presentation of π given by

ha, b|aba−3b2a−1b−1a3b−2_i. (4)

Since the relator in (4) satisfies aba−3b2a−1b−1a3b−2= a[ba−3b2, a−1b]a−1, this second presentation is

actually equivalent to

ha, b|[ba−3b2, a−1b]_i. (5)

An isomorphism between (2) and (4) is given by the changes of generators

(x, y) = (ab−1, ab−1a) and (a, b) = (x−1y, x−2y). (6)

Definition 1. The geometric representation of π is the morphism ρgeom: π−→ PSL(2, C) defined by

ρgeom(a) = u−1w1 =



i _−i

−1 − i 1



and ρgeom(b) = u−2w1 =−1 + 2i −2i

−1 − i 1



. (7)

The geometric representation is thus a discrete and faithful representation of π in PSL(2,C) with

image Γ. The above matrices for a and b are obtained by identifying respectively x and y to u and w1

2.3 Stabilizers of the vertices and peripheral curves

There are two orbits of vertices modulo the identifications in the octahedron_{O, the one of ∞ and the}

one of 0. It is a simple exercise using the face identifications to verify that the stabilisers of these two

points are respectively Γ∞ =hu, t2i, and Γ0 =hw1, uw1−1u−1t

−1 2 u−1w

−1

1 ui. The second generator for

Γ0 is the projective transformation associated to



1 0

2_{− 2i 1}

 . We express now these stabilisers in terms of a and b.

Proposition 2. The stabilizers of 0 and _{∞ in Γ are the images by ρ}geom of the two subgroups

of π respectively given by hab−1_{a, s}

0i and hab−1, s∞i, where s0 = [a, b−1]a−1b2a−3[b, a] and s∞ =

b−1a3b−1a−1.

Remark 1. SnapPy provides the following generators for the first homology of the boundary tori

of the Whitehead link complement using the presentation (5) (mi stands for meridian, and li for

longitude):

m1 = a−2b, l1 = a−2bab−2ab, m2 = b−1a, l2= b−1ab−1aba−3ba. (8)

By a direct computation, on verifies that

ρgeom(am1a−1) = w1 ρgeom(al1a−1) = w1−2s0

ρgeom(am2a−1) = x ρgeom(al2a−1) = t−1₂ u2

We see therefore that m1 and l1 correspond to the cusp of W associated to the (orbit of) 0, and that

m2 and l2 correspond to the one associated to the (orbit of)∞.

3

_{The SL(3,C)-character variety}

3.1 Generalities

Definition 2. Let G be a finitely generated group. The representation variety of G in SL(3,C) is Hom(G, SL(3, C).

Its GIT quotient

Hom(G, SL(3, C))//SL(3, C)

is called the SL(3,C)-character variety, denoted by χ3(G).

We refer the reader to [Sik12, Heu16] for classical definitions about representation and character

varieties and associated objects. A remark is important for our purposes: if G0 is a quotient of G,

then there is a natural map:

Hom(G0, SL(3, C)) ,→ Hom(G, SL(3, C)).

Indeed, a representation ρ0 : G0_{→ SL(3, C) is naturally promoted to a representation}

ρ : G  G0 ρ

0

−→ SL(3, C).

As the projection G  G0 is surjective, this map is injective and moreover two representation ρ0 and

¯

ρ0 in Hom(G0, SL(3, C)) are conjugate if and only if their associated ρ and ¯ρ in Hom(G, SL(3, C)) are

conjugate.

As a consequence of this discussion, we obtain

3.2 A quotient of π

We denote by π0 the quotient of π defined by the extra relations a3 = b3 = 1. More precisely, a

presentation for π0 is given by π0 =_{ha, b|a}3, b3, [ba−3b2, a−1b]_{i. Clearly, the group π}0 is isomorphic to

Z3? Z3: the last relation is a consequence of the first two. Moreover, π0 is the fundamental group of

a double Dehn surgery on the Whitehead link:

Proposition 4. The group π0 ' Z3? Z3 is isomorphic to the quotient of π defined by the two relations

m3₁l₁−1 and m3₂l−1₂ .

The proof of Proposition 4 is a direct verification from the definition of li and mi given in Remark

1: the two conditions m3₁l₁−1= m3₂l−1₂ = 1 imply that a3 = b3 = 1. In terms of Dehn surgery, π0 is the

fundamental group of the double Dehn surgery of slopes (_{−3, −3) on the Whitehead link. This double}

Dehn surgery is not hyperbolic: this may be verified using SnapPy (see Section 1 of the companion Sage notebook to this paper [Gui]). More precisely, it can be seen to be the connected sum of two lens spaces (see [MP06, Table 2]).

3.3 A lower bound for dim(X0)

We are now going to describe the SL(3,C)-character variety of π0. To this end, we use Lawton’s

theorem on the SL(3,C)-character variety of the rank two free group F2 [Law10].

Theorem 5 (Lawton [Law10]). The map ψ defined by

SL(3, C)× SL(3, C) → C8

(A, B) _{7→ (trA, trB, trAB, trA}−1B, trA−1, trB−1, trA−1B−1, trAB−1)

is onto C8 and descends to a (double) branched cover ψ : χ3(F2)−→ C8.

The theorem in Lawton’s work is more precise and gives an explicit polynomial in 9 variables

defining χ3(F2) as a hypersurface in C9 covering C8. Namely, the above double cover corresponds to

the fact that the traces of the nine words A, B, AB, A−1B, A−1, B−1, A−1B−1, AB−1 and [A, B]

sastify a relation of the form

(tr[A, B])2_{− S · tr[A, B] + P = 0,} (9)

where S and P are polynomials in the traces of the first eight above words. In other words, once the

traces of A, B, AB, A−1B, A−1, B−1, A−1B−1, AB−1 are fixed, the trace of [A, B] is determined

up to the choice of a root of (9). We provide the precise values of S and P in Section 7 of the Sage notebook [Gui], they may also be found in Lawton’s [Law10], or in [Wil16].

We can now give an alternate definition of the set X0 considered in the introduction:

Definition 3. Let X0 ⊂ χ3(F2) be the inverse image by ψ of the subspace

V =_{{(0, 0, z}1, z2, 0, 0, z3, z4), zi ∈ C}.

By Proposition 3, the sequence of quotients F2  π  π0 gives rise to a sequence of inclusions:

χ3(π0)⊂ χ3(π)⊂ χ3(F2).

With these inclusions in mind, we see that X0 is actually included in χ3(π0). We can even be more

specific:

Proposition 6. The set X0 is an irreducible Zariski closed subset of χ3(π0). Its dimension is at least

Proof. First, X0 is included in χ3(π0). Indeed, the condition ψ(A, B)∈ V rewrites

trA = trA−1 = trB = trB−1= .0

This implies that both A and B have order three. Indeed, the characteristic polynomial of a matrix

M _{∈SL(3,C) is equal to X}3_{− trMX}2+ trM−1X_{− 1 and thus if trM = trM}−1= 0 we have M3 = Id.

By construction, X0 is Zariski closed. Its irreducibility is not hard to verify using the explicit form

of the branched 2-cover of Theorem 5 given in [Law10]. For example, using the parametrisation given in Section 5, it is easily seen that the double cover is indeed a branched one and not the union of two distinct sheets: the discriminant of the quadratic equation defining the double cover is not a square. For a direct and precise proof, see also [Aco16, Section 4.1]. Eventually the dimension follows from

the fact it is the pull-back of C4 by ψ.

Remark 2. In fact X0 is a component of χ3(π0) and the only one of positive dimension. Moreover

it contains every irreducible representation of π0. The first part of this statement is a consequence of

Theorem 1. The character variety χ3(π0), with a focus on real points, has been studied in details in

[Aco16].

3.4 An upper bound for dim(X0)

We give in this section an upper bound on the dimension of the component of χ3(π) containing X0

by looking at a specific point to determine the Zariski tangent space. To this end, we consider the following elements of SL(3, C): S =    1 √ 3−i√5 2 −1 −√3−i√5 2 −1 0 −1 0 0    et T =    1 √ 3+i√5 2 −1 √ 3−i√5 2 −1 0 −1 0 0   

The matrices S and T have order three and satisfy tr(S) = tr(S−1) = tr(T ) = tr(T−1) = 0. We define

a point [ρ0] in the character variety of π by setting:

ρ0(a) = S ρ0(b) = T. (10)

By construction, [ρ0] belongs to X0. The key step in the proof of Theorem 1 is the following

Proposition 7. The Zariski tangent space to χ3(π) at [ρ0] has dimension 4.

We postpone the proof of Proposition 7 to the next section and proceed with the proof of Theorem

1. Recall that we need to prove that X0 is an algebraic component of χ3(π) containing [ρ0].

Proof of Theorem 1. Let X be an algebraic component of χ3(π) containing X0. The class [ρ0] belongs

to X. By Proposition 7, the dimension of X is at most 4: it is bounded above by the dimension of

any Zariski tangent space. But, by Proposition 6, X0 ⊂ X is a Zariski closed subset of dimension 4.

Hence, X = X0 and the theorem is proved.

4

Decorated representations and the deformation variety

We are going to compute the dimension of the Zariski tangent space of χ3(π) at [ρ0], in order to prove

Proposition 7. To this end, we will use a variation of the character variety – called decorated character variety – and a specific set of coordinates on it – the deformation variety. The deformation variety is

well-adapted to explicit computation. The equations defining this variety may be reconstructed using SnapPy’s command gluing equations pgl [CDW].

The tools hereafter presented are suitable for character varieties with target group the quotient PGL(3, C) rather than SL(3, C). This will not be a problem, as we use these tools for computing local dimension around a point which belongs to both character varieties. Indeed, if ρ is a representation of π in SL(3, C), the local dimensions at [ρ] of the character varieties for SL(3, C) and PGL(3, C) are the same.

4.1 Decorated representations

We first recall basic definitions. More details can be found in [BFG+13].

Definition 4. A flag of P(C3_{) is a pair ([x], [f ]) in P(C}3_{)× P((C}3₎∨_{) such that f (x) = 0. We denote}

by Fl3 the set of flags of P(C3).

Geometrically a flag is a pair formed by a point in P(C3) and a projective line containing it.

Definition 5. Let Γ be the fundamental group of a finite volume, cusped hyperbolic manifold M , let

P ⊂ H3

R be the set of parabolic fixed point of Γ and let ρ be a representation ρ : Γ−→ PGL(3, C). A

decoration of ρ is a map φ :_{P −→ Fl}3 which is (Γ, ρ)-equivariant. A pair (ρ, φ) is called a decorated

representation.

Definition 6. The decorated representation variety is

DecHom(Γ) ={(ρ, φ), ρ ∈ Hom(Γ, PGL(3, C)), φ is a decoration of ρ}.

The decorated character variety is the GIT quotient

Decχ3 = DecHom(Γ)//PGL(3, C)

The precise links between the different versions of representation and character varieties are

de-scribed in details in the introduction of [FGK+_{16]. It should be noted that for a given generic}

representation ρ : Γ → PGL(3,C), there exists only a finite number of possible decorations. Indeed,

if p _{∈ P is the fixed point of a parabolic element γ ∈ Γ, it should be mapped by φ to a flag that is}

invariant for ρ(γ). By equivariance, the map φ is completely determined by its values on a choice of

representatives of the orbits of Γ on P. As there is a finite number of cusps, and that each generic

element in SL(3,C) preserves a finite number of flags, the number of possible φ for a given ρ is finite.

For our purposes, we chose the point [ρ0] in χ3(π) (ρ0 is described in Section 3.4).

Proposition 8. The representation ρ0 admits a unique decoration.

Proof. The (hyperbolic) Whitehead link complement has two cusps, which are represented by the

sta-bilisers of 0 and_{∞ (see Proposition 2). Therefore, the equivariance property implies that a decoration}

φ of ρ0 is completely determined by the images φ(0) and φ(∞). The images by ρ0 of the stabilisers of

0 and_{∞ are respectively the cyclic groups hST}−1S_{i and hST}−1_{i. Indeed, the images of the stabilisers}

of 0 and_{∞ by ρ}0 are respectivelyhρ0(ab−1a), ρ0(s0)i and hρ0(ab−1), ρ0(s∞)i (this follows directly from

Proposition 2). The images by ρ0 of s0 and s∞ are ST−1S and T S−1 : this is a direct verification

using S3= T3= 1.

Now, the two maps ST−1 and ST−1S are regular unipotent, and thus each of them has only one

invariant flag. Therefore ρ0 can only be decorated in one way : φ must map 0 to the invariant flag of

The invariant flags of ST−1 and ST−1S (as well as those of various elements in the group) are

made explicit in Table 2. A consequence of Proposition 8 is that around [ρ0], the decorated character

variety is a finite ramified cover of the character variety (see also [Gui15]). As a consequence, the

local dimension around [ρ0] can be equivalently computed at the level of χ3(π) or of the decorated

character variety.

4.2 Using a triangulation: the deformation variety

A configuration of ordered points in a projective space P(V ) is said to be in general position when

they are all distinct and no three points are contained in the same line. This notion applies to

configurations of projective lines by duality. A configuration of n flags (([x1], [f1],· · · , ([xn], [fn]))) is

in general position whenever the n points ([xi])ni=1 are in general position and the forms [fj] satisfy

fj(xi)6= 0 when i 6= j.

Definition 7. We call tetrahedon of flags in P(C3) any 4-tuple of flags in general position.

We briefly recall now the definitions of the main projective invariants we are going to use as well as

the relations among them. We refer the reader to [BFG14] for more details. Let T = (F1, F2, F3, F4)

be a tetrahedron of flags.

1. Triple ratio. Let ijk
be a face of T (oriented accordingly to the orientation of the tetrahedron) of flags in general position. Its triple ratio is the quantity

zijk =

fi(xj)fj(xk)fk(xi)

fi(xk)fj(xi)fk(xj)

. (11)

2. Cross-ratio. For each oriented edge (ij), we define k and l in such a way that the permutation

(1, 2, 3, 4) _{7−→ (i, j, k, l) is even. Viewing the set of lines through [x}i] as a projective line, we

associate to (ij) the cross-ratio

zij =ker(fi), (xixj), (xixk), (xixl). (12)

These invariants can be thought of as decorating tetrahedra as shown in Figure 3 : to each face is associated a triple ratio, and to each edge are associated a pair of cross-ratios. Namely, the two cross

ratios zij and zji are associated to the edge (ij).

The above projective invariants are tied by the following internal relations:

zijk= 1 zikj , zijk =−zilzjlzkl zik = 1 1_{− z}ij , zijzikzil =−1. (IR)

In particular, the triple ratio can be expressed purely in terms of cross ratios. These invariants can be used to parametrise the set of tetrahedra of flags :

Proposition 9 (Proposition 2.10 of [BFG14]). A tetrahedron of flags is uniquely determined up to

i j k l zij zji zki zil zik zijk zilj zikl

Figure 3: The z-coordinates for a tetrahedron.

Remark that 0 and 1 are forbidden values (as is _{∞) because we assume the flags to be in general}

position: hence every cross-ratio is the cross-ratio of four different points.

Let now M be an ideally triangulated cusped hyperbolic 3-manifold. Denote by ν the number

of tetrahedra and by (∆µ)νµ=1 the family of tetrahedra triangulating M . We construct a decorated

representation of Γ = π1(M ) by turning each tetrahedron into a tetrahedron of flags and compute

the (decorated) monodromy of their gluing. We only need to ensure that we may glue the tetrahedra together in a consistent way :

1. whenever two tetrahedra ∆ and ∆0 are glued together along faces T ⊂ ∆ and T0 _{⊂ ∆}0_{, T and}

T0 should have the same shape, that is the same triple ratio up to inversion,

2. around each edge of the triangulation, the monodromy should be the identity.

These gluing conditions are described in details in [BFG+_{13, Section 2.3]. They give an equation for}

each face of the triangulation and two for each edge, which are respectively called the face equations and the edge equations.

Definition 8. The deformation variety of M , denoted Defor3(M ), is the subset of C12ν given by the

(zij(∆µ))06i6=j63, 16µ6ν verifying the internal relation (IR) for each tetrahedron Tµ together with the

face and edge equations.

In the case of the Whitehead Link Complement, the gluing equations are the 16 monomial equations displayed in Table 1. Hence, the deformation variety of the Whitehead Link Complement is the affine

algebraic subset of C48 defined by the 32 internal relations (IR) and 16 gluing equations of Table 1.

The holonomy map, as defined in [BFG14, GGZ15], is a well-defined map from the deformation variety

to the character variety χ3(π).

4.3 Finding [ρ0] in the deformation variety

The specific representation ρ0 we consider is defined by ρ0(a) = S and ρ0(b) = T where S and T are

the order three elements in SL(3,C) given in Section 3.4. We have seen in Proposition 8 that ρ0 admits

Face equations Edge equations z41(∆0)z31(∆0)z21(∆0)z41(∆1)z31(∆1)z21(∆1) = 1 z43(∆0)z34(∆1)z34(∆2)z34(∆3) = 1 z42(∆0)z32(∆0)z12(∆0)z42(∆2)z32(∆2)z12(∆2) = 1 z34(∆0)z43(∆1)z43(∆2)z43(∆3) = 1 z41(∆2)z31(∆2)z21(∆2)z42(∆3)z32(∆3)z12(∆3) = 1 z21(∆0)z12(∆1)z21(∆2)z21(∆3) = 1 z34(∆2)z24(∆2)z14(∆2)z43(∆3)z23(∆3)z13(∆3) = 1 z12(∆0)z21(∆1)z12(∆2)z12(∆3) = 1 z43(∆0)z23(∆0)z13(∆0)z34(∆3)z24(∆3)z14(∆3) = 1 z24(∆0)z23(∆0)z13(∆0)z24(∆1)z23(∆1)z13(∆1)z14(∆2)z23(∆3) = 1 z34(∆0)z24(∆0)z14(∆0)z34(∆1)z24(∆1)z14(∆1) = 1 z42(∆0)z31(∆0)z32(∆0)z42(∆1)z31(∆1)z32(∆1)z41(∆2)z32(∆3) = 1 z42(∆1)z32(∆1)z12(∆1)z41(∆3)z31(∆3)z21(∆3) = 1 z41(∆0)z41(∆1)z42(∆2)z31(∆2)z32(∆2)z42(∆3)z41(∆3)z31(∆3) = 1 z43(∆1)z23(∆1)z13(∆1)z43(∆2)z23(∆2)z13(∆2) = 1 z14(∆0)z14(∆1)z24(∆2)z23(∆2)z13(∆2)z24(∆3)z13(∆3)z14(∆3) = 1

Table 1: Gluing equations for the Whitehead Link Complement

of the octahedron described in Section 2.1 (see also the companion notebook [Gui, Section 4]). Note

that ρ0 maps every stabiliser of a vertex of the octahedron to a cyclic group. The flag associated to

this vertex is in fact invariant under the image by ρ0 of the stabiliser.

As a result, each tetrahedron (∆ν)16ν64 is decorated by four flags. For instance, the tetrahedron

∆0 = (i, 0,−1+i₂ ,∞) is decorated by (Fi, F0, F−1+i

2 , F∞) and the other three tetrahedra are decorated

in a similar way (the tetrahedra are listed in Section 2.2). It is a simple calculation to compute the cross-ratios associated to these flags as explained in Section 4.2. Table 3 displays, for each tetrahedron,

the values of coordinates z12, z21, z34, z43 (see also the companion notebook [Gui, Section 5]). The

values of the other coordinates can be deduced from them using the internal relations (IR).

Remark 3. Note the high degree of symmetry of the considered decorated representation: the tetra-hedra are all the same up to the action of SL(3, C) and a renumbering of the vertices.

4.4 Computation of the Zariski tangent space: proof of Proposition 7.

The deformation variety is the (algebraic) set of all tuples of 48 complex numbers satisfying both the

internal relations and the gluing equations (compare to [BFG+13, Section 3]). In other words it is the

intersection of the inverse images

IR−1(1, . . . , 1)_{∩ GR}−1(1, . . . , 1),

where the two maps IR and GR are defined by

• IR : (C \ {0, 1})48 _{→ (C}∗₎32 _{is the map representing the internal relations (IR): it sends a}

collection (zij(∆ν)) (for every half-edge ij and 1 6 ν 6 4) to the collection of 32 complex

numbers given by:

(−zij(∆ν)zik(∆ν)zil(∆ν), zik(∆ν)(1− zij(∆ν))).

• GE : (C \ {0, 1})48_{→ (C}∗₎16 _{is the collection of left-hand sides in the gluing equations of table}

1.

We denote by (IR, GE) the map (C\ {0, 1})48_{→ (C}∗₎48 _{given by the previous two maps. The Zariski}

tangent space to the deformation variety is the kernel of the tangent map to (IR, GE). As those maps are mostly monomial, we choose to write their tangent maps in the following basis of tangent spaces

Vertex Generator of its stabilizer in the image of ρ0 Invariant flag ∞ ST−1 F∞: h₁ 0 0 i , [0, 0, 1] 0 ST−1S F0:  ₁ −3 √ 3+i√5 4 −1  , [1,3 √ 3−i√5 4 ,−1] i S−1T−1 Fi : " ₁ − √ 3+i√5 4 −1+i√15 4 # , [1+i √ 15 4 , √ 3−i√5 4 ,−1] −1 + i T−1ST−1 F−1+i :  ₁ 3√3−i√5 4 −1  , [1,−3√3+i√5 4 ,−1] −i T−1S−1 F−i: " ₁ √ 3−i√5 4 −1+i√15 4 # , [1−i √ 15 4 ,− √ 3+i√5 4 ,−1] −1+i 2 T S F−1+i₂ : h₀ 0 1 i , [1, 0, 0]

Table 2: The unique decoration of ρ0

at the source and target. We take, for each coordinate z, the vector field z_∂z∂. In these basis, the

tangent map to a function φ : (C∗)48_{→ C}∗ _{has entries of the form}

z φ

∂φ

∂z.

This follows from the following elementary lemma:

Lemma 10. Let f : C∗ → C∗ _{be a differentiable function. In the basis z} ∂

∂z, the tangent map T f at

a_{∈ C}∗ has coordinate _{f (a)}a ∂f_∂z.

Proof. In usual coordinates, by definition of partial derivative, the tangent map Taf at a point a maps

v _{∈ T}aC∗ to w = ∂f

∂zv∈ Tf (a)C

∗_.

The basis change from _∂z∂ to z_∂z∂ in both tangent spaces transforms v into u = v_a and w into _{f (a)}w .

Hence, in new basis, u is sent to w f (a) = 1 f (a) ∂f ∂zv = a f (a) ∂f ∂zu.

Tetrahedron z12 z21 z34 z43 ∆0 7+i √ 15 4 −1−i√15 8 7−i√15 4 −1+i√15 8 ∆1 −1+i √ 15 8 7−i√15 4 −1−i√15 8 7+i√15 4 ∆2 7−i √ 15 4 −1+i√15 8 −1−i√15 8 7+i√15 4 ∆3 −1−i √ 15 8 7+i√15 4 −1+i√15 8 7−i√15 4

Table 3: Coordinates for [ρ0] in the deformation variety

We will apply this lemma to each coordinate of the map (IR, GE) to obtain a matrix for the

tangent map to (IR, GE) at the point [ρ0]. This matrix, denoted J , has size 48× 48 and is depicted

in Table 4 (see Remark 4 below). To construct J , we have to deal with two kinds of functions, depending on the equations that form the the maps IR and GE : monomial maps or maps of the form

zik(∆ν)(1− zij(∆ν)).

• If f is a monomial map, its tangent map has integer entries equal to the exponent of the relative

variable. As an example, this formula applied to the map _−z12(∆0)z13(∆0)z14(∆0) gives all

entries equal to 0 except for those associated to the variables z12(∆0), z13(∆0) and z14(∆0)

which give three entries equal to 1. The same phenomenon appears for each of the first sixteen rows of the matrix J . The gluing equations are also monomials (see Table 1), but involve more variables. These correspond to lines 33 to 48 of the matrix J , that have all their coefficients equal to 0 except for 4, 6 or 8 ones that are equal to 1.

• if f has the form1 _z

ik(1− zij), its tangent map has every entry 0 except the ones corresponding

to zij and zik. Those two are respectively −_1−zzij_ij and 1. Note that, at a point verifying the

internal relations, we have the additional relation ₋ zij

1−zij = zil (see also the computation in

[BFG+13, Section 5], especially Lemma 5.3). Hence the entries for such a map are 0, 1 or zil.

Those appear in rows 17 to 32 of the matrix J displayed in Table 4.

We see thus that J has entries either integer or of the form zil(∆ν). Note moreover that the last 16 rows,

corresponding to the gluing equations, can be accessed directly by SnapPy [CDW] under SageMath [Dev16]: it is the Neumann-Zagier datum. This part of J is directly given by the commands:

import snappy;

Triangulation("5^2_1").gluing_equations_pgl(3,equation_type=’non_peripheral’).matrix

The next step is to compute the kernel of J . As all entries are in the number field Q[i,√3,√5], a

computer algebra system such as Sage computes it exactly. As a result, the dimension of this kernel is 4 (see Section 6 of the Sage notebook [Gui]). We deduce that the dimension of the Zariski tangent

space at the decoration of [ρ0] to the deformation variety is 4.

1

Remark 4. To write the matrix J , we choose the same order on the variables zij(∆ν) as SnapPy

does. As the recise order it is not very enlightning, we omit this discussion here. A change of order on the variables amounts to a permutation of the columns of J , which does not affect its kernel.

Note that at [ρ0], the two subgroups generated by the pairs (ρ0(li), ρ0(mi)) for i = 1, 2 are regular

unipotent: there is only one invariant flag for each one. As noted before, it implies that, locally the holonomy map between the deformation variety and the actual character variety is a finite ramified covering. This concludes the proof of Proposition 7: the Zariski tangent space to the character variety

at [ρ0] also has dimension 4.

Row 1: 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 0 0 0 Row 16: 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 1 Row 17: 1 0 ¯x 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 y¯ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 x 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 y 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 ¯x 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 ¯y 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 x 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 y 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 x 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 y 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 ¯x 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 y¯ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 y 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 x 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 ¯y 0 0 0 Row 32: 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 ¯x Row 33: 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 1 0 1 0 0 0 0 0 0 0 0 1 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 0 1 1 0 0 0 0 0 0 0 1 0 0 1 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 1 0 0 1 1 0 0 0 0 0 0 0 1 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 1 0 1 0 0 0 0 0 0 0 0 1 0 0 1 1 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 1 0 1 0 1 0 0 0 0 0 0 0 1 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 0 0 0 0 1 0 1 0 1 0 0 0 0 0 0 0 1 0 1 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 1 0 1 0 0 0 0 0 0 1 0 0 0 1 0 0 0 1 0 0 Row 48: 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 1 0 0 0 1 0 0 0 1 1 0 0 0 0 0 0 0 1 0 1 0

Table 4: The matrix J , where x = 9+i

√ 15 8 and y =− 3+i√15 4 .

5

A parametrisation of

X

We describe in this section a parametrisation of a Zariski open subset of X0 by actual matrices. More

precisely, given four generic complex numbers z1, z2, z3 and z4 we are going to provide two pairs

(A, B) of regular order three elements in SL(3,C) such that

• A and B have no common eigenvector in C3_,

• A and B satisfy

Recall that the traces of A, B and their inverses are zero as they are regular order three element. The genericity condition will be made explicit in Proposition 11.

We know from Lawton’s theorem that X0 is a double cover over C4. Parameters on C4 are given

by the four trace parameter z1, z2, z3, z4. To describe this double cover, define the quantity

∆ = z₁2z₃2− 2z1z2z3z4+ z22z42− 4z13− 4z23− 4z33− 4z43+ 18z1z3+ 18z2z4− 27,

which is the discriminant of the trace equation (9) in Theorem 5, in the case where the traces of A, B and their inverses vanish. We denote by δ a square root of ∆. Let j be a non trivial cube root of 1

and k = Q[j](z1, z2, z3, z4).

Proposition 11. Let (a, b, c, d) be the following elements in k[δ]:

4a = z1z3− z2z4+ 6jz1+ 6j 2_z 3+ 9 + δ jz1+ z2+ j2z3+ z4+ 3 4d = z1z3− z2z4+ 6j 2_z 1+ 6jz3+ 9− δ j2_z 1+ z2+ jz3+ z4+ 3 b = z1− z2− j 2_z 3+ j2z4+ 3(j2− 1) z1+ z2+ j2z3+ j2z4+ 3j a + (j_{− 1)}z1+ jz2+ jz3+ z4+ 3j 2 z1+ z2+ j2z3+ j2z4+ 3j c = z1+ jz2− jz3− z4+ 3(j− 1) z1+ jz2+ jz3+ z4+ 3j2 d + (j2_{− 1)}z1+ z2+ j 2_z 3+ j2z4+ 3j z1+ jz2+ jz3+ z4+ 3j2 .

Then for any 4-tuple of complex numbers (z1, z2, z3, z4) such that a, b c and d are well-defined, any

pair (A, B) of order three regular elements of SL(3,C), satisfying (13) and such that the j2-eigenline

for A is different from the j-eigenline for B, is conjugate in SL(3,C) to one of the two pairs defined by A =   j 0 0 j2 1 0 b + a 2ja j2   and B =   j 2j2d c + d 0 1 j 0 0 j2  . (14)

Proof. A direct computation of the traces of AB, A−1B, A−1B−1 and AB−1 with the given values

leads to a verification of our parametrization [Gui, Section 7]. We now indicate how to obtain these values.

Recall first that regular order three elements in SL(3,C) have eigenvalue spectrum {1, j, j2_{}. First,}

one may conjugate the pair (A, B) so that A and B are respectively lower and upper triangle, with

eigenvalues organised as in (14). This amounts to choosing a basis of C3 of the form (vB, v, vA) where

vA(resp. vB) is a j2-eigenvector for A (resp. a j-eigenvector for B), and v is a non zero vector in the

intersection VA∩ VB, where VA(resp. VB) is spanned by vAand a 1-eigenvector of A (resp. vB and a

1 eigenvector for B). Conjugating by a diagonal matrix allows to bring off-diagonal coefficients equal

to j2 in A and to j in B as shown in (14).

We need now to determine a, b, c, and d from (13). These four conditions correspond to the

following system of equations.

Σ
           (a + b)(d + c) + 2aj2+ 2jd = z1 (a− b)(d + c) − 2a − 2d + 3 = z2 (a_{− b)(d − c) + 2ja + 2j}2d = z3 (a + b)(d_{− c) − 2a − 2d + 3 = z}4 (15) (16) (17) (18)

This system is relatively easy to solve using a computer and, for instance, Gr¨obner bases. However, it is also solvable by hand, and we indicate now how to do it. Before going any further, let us observe that conjugating the pair (A, B) by the matrix

  0 0 1 0 1 0 1 0 0  

amounts to do following exchanges in Σ:

a_{←→ d, b ←→ c, j ←→ j}2. (19)

More precisely, the left-hand sides of (15) and (17) are preserved by these changes, whereas the left-hand sides of (16) and (18) are exchanged.

Next, we compute linear combinations of the above equations, and obtain the following equivalent system: Σ0
           (15) + (16) + (17) + (18) : 4ad_{− 6a − 6d − z}1− z2− z3− z4+ 6 = 0 (15)_{− (16) + (17) − (18) :} 4bc + 2a + 2d_{− z}1+ z2− z3+ z4− 6 = 0 (15)_{− (17) :} 2ac + 2bd_{− 4aj + 4dj − 2a + 2d − z}1+ z3 = 0 (16)_{− (18) :} 2ac_{− 2bd − z}2+ z4 = 0 (20) (21) (22) (23) To obtain the value of a annonced in the statement, we proceed ad follows.

• The two equations (22) and (23) are linear in b and c. We solve them to obtain expressions of b and c in terms of a and d.

• Plugging these expressions of b and c in (21) and taking numerator, we obtain an equation that

relates a and d and involves the monomials a2d, ad2, a2, d2, ad, a and d. This equation can by

simplified by observing that the product ad can be expressed as an affine function of a and d using (20). Doing so, most of the monomials simplify and we obtain a linear relation between a and d. This yields an expression of d as a function of a, which can be inserted back in (20). We obtain this way a quadratic equation in a, which is:

0 =4(jz1+ z2+ j2z3+ z4+ 3)a2− 2(6jz1+ 6j2z3+ z1z3− z2z4+ 9)a

+ 9 + 6jz1− 3z2+ 6j2z3− 3z4+ j2z21+ z22+ jz32+ z24

− jz1z2+ 2z1z3− jz1z4− j2z2z3− z2z4− j2z3z4

(24)

The discriminant of this quadratic equation is ∆, and we obtain in turn two possible values for a, corresponding to the two square roots of ∆. We obtain in turn the value of d given in the statement.

Note that d is obtained from a by the exchanges j _{←→ j}2 and z2 ←→ z4. This correspond to the

symmetry of the system (Σ) given in (19).

As observed above, knowing the values of a and d gives us the values of b and c. However,

the expressions obtained by solving (22) and (23) are not exactly those given in the statement, and simplifying them is quite intricate. The following strategy gives a way to determine b and c more directly. First of all, we know from Lawton’s theorem and our determination of a and d that b belongs

to k[δ] (recall that k = Q[j](z1, z2, z3, z4)). Hence, we may look for it under the form b = aP + Q

where P and Q belong to k. We use this form in the equation (22)_{−(23), and also plug the values of}

coefficient of δ and the remaining part, we find 2 linear equations in P and Q. Solving those equations leads to the given value for b. The value for c can be obtained using the symmetries of Σ given in (19).

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